Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 99.2%

Time: 5.8s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l, om, kx, ky)
use fmin_fmax_functions
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{l\_m + l\_m}{Om\_m}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 2:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\_m\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{t\_0 \cdot \mathsf{hypot}\left(\sin ky\_m, \sin kx\_m\right)}\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l_m l_m) Om_m)))
   (if (<=
        (sqrt
         (+
          1.0
          (*
           (pow (/ (* 2.0 l_m) Om_m) 2.0)
           (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
        2.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 ky_m)))) (* t_0 t_0) 1.0)))
        0.5)))
     (sqrt
      (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (* t_0 (hypot (sin ky_m) (sin kx_m))))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (l_m + l_m) / Om_m;
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 2.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * ky_m)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / (t_0 * hypot(sin(ky_m), sin(kx_m)))))));
	}
	return tmp;
}

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(l_m + l_m) / Om_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 2.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky_m)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / Float64(t_0 * hypot(sin(ky_m), sin(kx_m)))))));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l$95$m + l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[(t$95$0 * N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx$95$m], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 2

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   3. Applied rewrites 99.8%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      4. lift--.f64 99.5

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   6. Applied rewrites 99.5%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   if 2 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 96.7%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{2 \cdot \left(\frac{\ell}{Om} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}\right)}}\right)} \]
   3. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\left(2 \cdot \frac{\ell}{Om}\right) \cdot \color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{2 \cdot \ell}{Om} \cdot \sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{2 \cdot \ell}{Om} \cdot \color{blue}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)} \]

      4. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{2 \cdot \ell}{Om} \cdot \sqrt{\color{blue}{{\sin kx}^{2} + {\sin ky}^{2}}}}\right)} \]

      5. count-2-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}}\right)} \]

      7. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \sqrt{{\sin ky}^{2} + {\sin kx}^{2}}}\right)} \]

      8. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \sqrt{\sin ky \cdot \sin ky + {\sin kx}^{2}}}\right)} \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}\right)} \]

      10. lower-hypot.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)}\right)} \]

      11. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \mathsf{hypot}\left(\sin ky, \sin \color{blue}{kx}\right)}\right)} \]

      12. lift-sin.f64 98.8

          \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\frac{\ell + \ell}{Om} \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right)}\right)} \]

   4. Applied rewrites 98.8%

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\frac{\ell + \ell}{Om} \cdot \mathsf{hypot}\left(\sin ky, \sin kx\right)}}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{l\_m + l\_m}{Om\_m}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 2:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\_m\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.25, \frac{Om\_m}{l\_m} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky\_m, \sin kx\_m\right)}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l_m l_m) Om_m)))
   (if (<=
        (sqrt
         (+
          1.0
          (*
           (pow (/ (* 2.0 l_m) Om_m) 2.0)
           (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
        2.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 ky_m)))) (* t_0 t_0) 1.0)))
        0.5)))
     (sqrt
      (fma 0.25 (* (/ Om_m l_m) (/ 1.0 (hypot (sin ky_m) (sin kx_m)))) 0.5)))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (l_m + l_m) / Om_m;
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 2.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * ky_m)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(fma(0.25, ((Om_m / l_m) * (1.0 / hypot(sin(ky_m), sin(kx_m)))), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(l_m + l_m) / Om_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 2.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky_m)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(fma(0.25, Float64(Float64(Om_m / l_m) * Float64(1.0 / hypot(sin(ky_m), sin(kx_m)))), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l$95$m + l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(0.25 * N[(N[(Om$95$m / l$95$m), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[ky$95$m], $MachinePrecision] ^ 2 + N[Sin[kx$95$m], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 2

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   3. Applied rewrites 99.8%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      4. lift--.f64 99.5

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   6. Applied rewrites 99.5%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   if 2 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 96.7%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} + \frac{1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{1}{4}} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) + \color{blue}{\frac{1}{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{1}{4}, \color{blue}{\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}, \frac{1}{2}\right)} \]

   4. Applied rewrites 98.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)}}\right)} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l_m) Om_m) 2.0)
        (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0))))))))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0)))))))));
}

Fortran

l_m =     private
Om_m =     private
kx_m =     private
ky_m =     private
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l_m, om_m, kx_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0)))))))))
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l_m) / Om_m), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0)))))))));
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l_m) / Om_m), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0)))))))))

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))))
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp = code(l_m, Om_m, kx_m, ky_m)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0)))))))));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
2. Add Preprocessing

Alternative 4: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{l\_m + l\_m}{Om\_m}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 400000000000:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\_m\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l_m l_m) Om_m)))
   (if (<=
        (sqrt
         (+
          1.0
          (*
           (pow (/ (* 2.0 l_m) Om_m) 2.0)
           (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
        400000000000.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 ky_m)))) (* t_0 t_0) 1.0)))
        0.5)))
     (sqrt 0.5))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (l_m + l_m) / Om_m;
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 400000000000.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * ky_m)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(l_m + l_m) / Om_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 400000000000.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky_m)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l$95$m + l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 400000000000.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 4e11

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   3. Applied rewrites 99.4%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5}} \]

   4. Taylor expanded in kx around 0

      \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      4. lift--.f64 98.6

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   6. Applied rewrites 98.6%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   if 4e11 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 96.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.2%

      \[\leadsto \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} t_0 := \frac{l\_m + l\_m}{Om\_m}\\ \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 2:\\ \;\;\;\;\sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\_m\right), t\_0 \cdot t\_0, 1\right)}} \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25, \frac{Om\_m}{l\_m \cdot \sin ky\_m}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (let* ((t_0 (/ (+ l_m l_m) Om_m)))
   (if (<=
        (sqrt
         (+
          1.0
          (*
           (pow (/ (* 2.0 l_m) Om_m) 2.0)
           (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
        2.0)
     (sqrt
      (+
       0.5
       (*
        (/ 1.0 (sqrt (fma (- 0.5 (* 0.5 (cos (* 2.0 kx_m)))) (* t_0 t_0) 1.0)))
        0.5)))
     (sqrt (fma -0.25 (/ Om_m (* l_m (sin ky_m))) 0.5)))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double t_0 = (l_m + l_m) / Om_m;
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 2.0) {
		tmp = sqrt((0.5 + ((1.0 / sqrt(fma((0.5 - (0.5 * cos((2.0 * kx_m)))), (t_0 * t_0), 1.0))) * 0.5)));
	} else {
		tmp = sqrt(fma(-0.25, (Om_m / (l_m * sin(ky_m))), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	t_0 = Float64(Float64(l_m + l_m) / Om_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 2.0)
		tmp = sqrt(Float64(0.5 + Float64(Float64(1.0 / sqrt(fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * kx_m)))), Float64(t_0 * t_0), 1.0))) * 0.5)));
	else
		tmp = sqrt(fma(-0.25, Float64(Om_m / Float64(l_m * sin(ky_m))), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := Block[{t$95$0 = N[(N[(l$95$m + l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], N[Sqrt[N[(0.5 + N[(N[(1.0 / N[Sqrt[N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * kx$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(-0.25 * N[(Om$95$m / N[(l$95$m * N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 2

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   3. Applied rewrites 99.8%

      \[\leadsto \sqrt{\color{blue}{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5}} \]

   4. Taylor expanded in ky around 0

      \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]
   5. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right), \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} + \frac{1}{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot kx\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot \frac{1}{2}} \]

      4. lift--.f64 99.0

         \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(0.5 - \color{blue}{0.5 \cdot \cos \left(2 \cdot kx\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   6. Applied rewrites 99.0%

      \[\leadsto \sqrt{0.5 + \frac{1}{\sqrt{\mathsf{fma}\left(\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)}, \frac{\ell + \ell}{Om} \cdot \frac{\ell + \ell}{Om}, 1\right)}} \cdot 0.5} \]

   if 2 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 96.7%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around -inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{-1}{4}} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{-1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) + \color{blue}{\frac{1}{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}, \frac{1}{2}\right)} \]

   4. Applied rewrites 97.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \frac{Om}{\color{blue}{\ell \cdot \sin ky}}, \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \frac{Om}{\ell \cdot \color{blue}{\sin ky}}, \frac{1}{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \frac{Om}{\ell \cdot \sin ky}, \frac{1}{2}\right)} \]

      3. lift-sin.f64 97.8

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.25, \frac{Om}{\ell \cdot \sin ky}, 0.5\right)} \]

   7. Applied rewrites 97.8%

      \[\leadsto \sqrt{\mathsf{fma}\left(-0.25, \frac{Om}{\color{blue}{\ell \cdot \sin ky}}, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 98.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-0.25, \frac{Om\_m}{l\_m \cdot \sin ky\_m}, 0.5\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (+
        1.0
        (*
         (pow (/ (* 2.0 l_m) Om_m) 2.0)
         (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
      2.0)
   1.0
   (sqrt (fma -0.25 (/ Om_m (* l_m (sin ky_m))) 0.5))))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt(fma(-0.25, (Om_m / (l_m * sin(ky_m))), 0.5));
	}
	return tmp;
}

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt(fma(-0.25, Float64(Om_m / Float64(l_m * sin(ky_m))), 0.5));
	end
	return tmp
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[Sqrt[N[(-0.25 * N[(Om$95$m / N[(l$95$m * N[Sin[ky$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 2

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2}} \cdot \sqrt{2} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot 2} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot 2} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      5. metadata-eval 99.0

         \[\leadsto 1 \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if 2 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 96.7%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around -inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} + \frac{-1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{-1}{4}} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \]

      2. +-commutative N/A

         \[\leadsto \sqrt{\frac{-1}{4} \cdot \left(\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right) + \color{blue}{\frac{1}{2}}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \color{blue}{\frac{Om}{\ell} \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}}, \frac{1}{2}\right)} \]

   4. Applied rewrites 97.5%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(-0.25, \frac{Om}{\ell} \cdot \frac{1}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}, 0.5\right)}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \frac{Om}{\color{blue}{\ell \cdot \sin ky}}, \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \frac{Om}{\ell \cdot \color{blue}{\sin ky}}, \frac{1}{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\frac{-1}{4}, \frac{Om}{\ell \cdot \sin ky}, \frac{1}{2}\right)} \]

      3. lift-sin.f64 97.8

         \[\leadsto \sqrt{\mathsf{fma}\left(-0.25, \frac{Om}{\ell \cdot \sin ky}, 0.5\right)} \]

   7. Applied rewrites 97.8%

      \[\leadsto \sqrt{\mathsf{fma}\left(-0.25, \frac{Om}{\color{blue}{\ell \cdot \sin ky}}, 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ \begin{array}{l} \mathbf{if}\;\sqrt{1 + {\left(\frac{2 \cdot l\_m}{Om\_m}\right)}^{2} \cdot \left({\sin kx\_m}^{2} + {\sin ky\_m}^{2}\right)} \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m)
 :precision binary64
 (if (<=
      (sqrt
       (+
        1.0
        (*
         (pow (/ (* 2.0 l_m) Om_m) 2.0)
         (+ (pow (sin kx_m) 2.0) (pow (sin ky_m) 2.0)))))
      2.0)
   1.0
   (sqrt 0.5)))

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (sqrt((1.0 + (pow(((2.0 * l_m) / Om_m), 2.0) * (pow(sin(kx_m), 2.0) + pow(sin(ky_m), 2.0))))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = sqrt(0.5);
	}
	return tmp;
}

Fortran

l_m =     private
Om_m =     private
kx_m =     private
ky_m =     private
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l_m, om_m, kx_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    real(8) :: tmp
    if (sqrt((1.0d0 + ((((2.0d0 * l_m) / om_m) ** 2.0d0) * ((sin(kx_m) ** 2.0d0) + (sin(ky_m) ** 2.0d0))))) <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = sqrt(0.5d0)
    end if
    code = tmp
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	double tmp;
	if (Math.sqrt((1.0 + (Math.pow(((2.0 * l_m) / Om_m), 2.0) * (Math.pow(Math.sin(kx_m), 2.0) + Math.pow(Math.sin(ky_m), 2.0))))) <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = Math.sqrt(0.5);
	}
	return tmp;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	tmp = 0
	if math.sqrt((1.0 + (math.pow(((2.0 * l_m) / Om_m), 2.0) * (math.pow(math.sin(kx_m), 2.0) + math.pow(math.sin(ky_m), 2.0))))) <= 2.0:
		tmp = 1.0
	else:
		tmp = math.sqrt(0.5)
	return tmp

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0
	if (sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l_m) / Om_m) ^ 2.0) * Float64((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	return tmp
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp_2 = code(l_m, Om_m, kx_m, ky_m)
	tmp = 0.0;
	if (sqrt((1.0 + ((((2.0 * l_m) / Om_m) ^ 2.0) * ((sin(kx_m) ^ 2.0) + (sin(ky_m) ^ 2.0))))) <= 2.0)
		tmp = 1.0;
	else
		tmp = sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := If[LessEqual[N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l$95$m), $MachinePrecision] / Om$95$m), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx$95$m], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky$95$m], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], 1.0, N[Sqrt[0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))) < 2

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2}} \cdot \sqrt{2} \]

      2. sqrt-unprod N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot 2} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot 2} \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{1} \]

      5. metadata-eval 99.0

         \[\leadsto 1 \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{1} \]

   if 2 < (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))

   1. Initial program 96.7%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

   2. Taylor expanded in l around inf

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.7%

      \[\leadsto \sqrt{\color{blue}{0.5}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 61.8% accurate, 142.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ Om_m = \left|Om\right| \\ kx_m = \left|kx\right| \\ ky_m = \left|ky\right| \\ [l_m, Om_m, kx_m, ky_m] = \mathsf{sort}([l_m, Om_m, kx_m, ky_m])\\ \\ 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
Om_m = (fabs.f64 Om)
kx_m = (fabs.f64 kx)
ky_m = (fabs.f64 ky)
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
(FPCore (l_m Om_m kx_m ky_m) :precision binary64 1.0)

C

l_m = fabs(l);
Om_m = fabs(Om);
kx_m = fabs(kx);
ky_m = fabs(ky);
assert(l_m < Om_m && Om_m < kx_m && kx_m < ky_m);
double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return 1.0;
}

Fortran

l_m =     private
Om_m =     private
kx_m =     private
ky_m =     private
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(l_m, om_m, kx_m, ky_m)
use fmin_fmax_functions
    real(8), intent (in) :: l_m
    real(8), intent (in) :: om_m
    real(8), intent (in) :: kx_m
    real(8), intent (in) :: ky_m
    code = 1.0d0
end function

Java

l_m = Math.abs(l);
Om_m = Math.abs(Om);
kx_m = Math.abs(kx);
ky_m = Math.abs(ky);
assert l_m < Om_m && Om_m < kx_m && kx_m < ky_m;
public static double code(double l_m, double Om_m, double kx_m, double ky_m) {
	return 1.0;
}

Python

l_m = math.fabs(l)
Om_m = math.fabs(Om)
kx_m = math.fabs(kx)
ky_m = math.fabs(ky)
[l_m, Om_m, kx_m, ky_m] = sort([l_m, Om_m, kx_m, ky_m])
def code(l_m, Om_m, kx_m, ky_m):
	return 1.0

Julia

l_m = abs(l)
Om_m = abs(Om)
kx_m = abs(kx)
ky_m = abs(ky)
l_m, Om_m, kx_m, ky_m = sort([l_m, Om_m, kx_m, ky_m])
function code(l_m, Om_m, kx_m, ky_m)
	return 1.0
end

MATLAB

l_m = abs(l);
Om_m = abs(Om);
kx_m = abs(kx);
ky_m = abs(ky);
l_m, Om_m, kx_m, ky_m = num2cell(sort([l_m, Om_m, kx_m, ky_m])){:}
function tmp = code(l_m, Om_m, kx_m, ky_m)
	tmp = 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
Om_m = N[Abs[Om], $MachinePrecision]
kx_m = N[Abs[kx], $MachinePrecision]
ky_m = N[Abs[ky], $MachinePrecision]
NOTE: l_m, Om_m, kx_m, and ky_m should be sorted in increasing order before calling this function.
code[l$95$m_, Om$95$m_, kx$95$m_, ky$95$m_] := 1.0

Derivation

1. Initial program 98.4%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]

2. Taylor expanded in l around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \sqrt{2}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{2}} \cdot \sqrt{2} \]

   2. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot 2} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot 2} \]

   4. metadata-eval N/A

      \[\leadsto \sqrt{1} \]

   5. metadata-eval 61.8

      \[\leadsto 1 \]

4. Applied rewrites 61.8%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))